Ordered Partition

I have this math problem that drives me absolutely nuts. I saw this same one at high school and again in college. Here it is:

In how many ways can 9 students be divided into three teams?

The answer to this question is

At high school, I asked my teacher why the answer being it is. He asked me to consider myself a coach. By so, I appoint a student for a team captain for each team and have each team captain choose two others for her team. When I said, "If I were the coach, why can't I choose them myself?" He was speechless.

Now I see this same problem again in a different book that gave the answer in this form:

I understand this part: , but I don't understand why it's multiplied by